Contrôle de la dynamique d un bateau

Transcription

1 Contrôle de la dynamique d un bateau Lionel Rosier Université Henri Poincaré Nancy 1 Un peu de contrôle à Clermont-Ferrand Juin 2011

2 Joint work with Olivier Glass, Université Paris-Dauphine

3 Control of the motion of a boat We consider a rigid body S R 2 with one axis of symmetry, surrounded by a fluid, and which is controlled by two fluid flows, a longitudinal one and a transversal one.

4 Aims We aim to control the position and velocity of the rigid body by the control inputs. System of dimension 3+3 with a PDE in the dynamics. Control living in R 2. No control objective for the fluid flow (exterior domain!!). Model for the motion of a boat with a longitudinal propeller, and a transversal one (thruster) in the framework of the theory of fluid-structure interaction problems. Rockets and planes also concerned.

5 Bowthruster

6 Longitudinal thruster

7 What is a fluid-structure interaction problem? Consider a rigid (or flexible) structure in touch with a fluid. The velocity of the fluid obeys Navier-Stokes (or Euler) equations in a variable domain The dynamics of the rigid structure is governed by Newton laws. Great role played by the pressure. Questions of interest: existence of (weak, strong, global) solutions of the system fluid+solid, uniqueness, long-time behavior, control, inverse problems, optimal design,...

8 Main difficulties 1. The systems describing the motions of the fluid and the solid are nonlinear and strongly coupled; e.g., the pressure of the fluid gives rise to a force and a torque applied to the solid, and the fluid domain changes when the solid is moving. 2. The fluid domain R N \ S(t) is an unknown function of time

9 Why to consider perfect fluids? 1. Euler equations provide a good model for the motion of boats or submarines in a reasonable time-scale. 2. Explicit computations may be performed with the aid of Complex Analysis when the flow is potential and 2D. 3. There is a natural choice for the boundary conditions u rel n = 0 for Euler equations. For Navier-Stokes flows, one often takes u rel = 0 4. The control theory of Euler flows is well understood (Coron, Glass).

10 System under investigation Ω(t) = R 2 \ S(t) Euler u t + (u )u + p = 0, x Ω(t) div u = 0, x Ω(t) u n = (h + r(x h) ) n + w(x, t), x Ω(t) lim x u(x, t) = 0 Newton m h (t) = p n dσ Ω(t) J r = (x h) p n dσ Ω(t) System supplemented with Initial Conditions, and with the value of the vorticity at the incoming flow (in Ω(t)) for the uniqueness

11 System in a frame linked to the solid After a change of variables and unknown functions, we obtain in Ω := R 2 \ S(0) v t + (v l ry ) v + rv + q = 0, y Ω div v = 0, y Ω v n = (l + ry ) n + lim v(y, t) = 0 y 1 j 2 m l (t) = q n dσ mrl Ω J r = qn y dσ Ω where l(t) := Q(θ(t)) 1 h (t), r(t) = θ (t). w j (t)χ j (y), y Ω

12 Potential flows Assuming that the initial vorticity and circulation are null ω 0 := curl u 0 0, Γ 0 := u 0 n dσ = 0 and that the vorticity at the inflow part of Ω is null ω(y, t) = 0 if w i (t)χ i (y) < 0 for some i = 1, 2 then the flow remains potential, i.e. v = φ where φ solves φ = 0 in Ω [0, T ] φ n = (l + ry ) n + w i (t)χ i (y) on Ω [0, T ] i=1,2 lim y φ(y) = 0 on [0, T ] Ω

13 Potential flows (continued) v = φ decomposed as φ = l i (t) ψ i (y) + r(t) ϕ(y) + w i (t) θ i (y) i=1,2 i=1,2 where the functions ϕ, ψ i and θ i are harmonic on Ω and fulfill the following boundary conditions on Ω ϕ n = y n, ψ i n = n i(y), θ i n = χ i(y) This gives the following expression for the pressure q = { i=1,2 l i ψ i + r ϕ + i=1,2 w i θ i + v 2 2 l v ry v} Plugging this expression in Newton s law yields a

14 Control system in finite dimension h = Ql J l = Cw + B(l, w) where Q = diag(q, 1), h = [h 1, h 2, θ] T, l = [l 1, l 2, r] T, w = [w 1, w 2 ] T is the control input, and m + ψ 1 n J = 0 m + ψ 2 n 2 ψ2 y n 0 ψ2 y n J + ϕy n C = θ 1 n θ 2 n 2 0 θ 2 y n = where = Ω and B(l, w) is bilinear in (l, w) c c 2 0 c 2

15 Toy problem w 2 = 0, h 2 = l 2 = 0 where (α, β, γ) := (m + Claims { h ( ) 1 = l 1 l 1 = αw 1 + βw 1l 1 + γw1 2 Ω ψ 1 n 1 ) 1 ( θ 1 n 1, χ 1 1 ψ 1, χ 1 1 θ 1 ) Ω Ω Ω If we add the equation w 1 = v 1 to ( ), the system with state (h 1, l 1, w 1 ) and input v 1 is NOT controllable! In general we cannot impose the condition w 1 (0) = w 1 (T ) = 0 when l 1 (0) = l 1 (T ) = 0 (i.e. fluid at rest at t = 0, T ). Actually we can do that if and only if γ + αβ = 0.

16 Proof of the claims Introduce z 1 := l 1 αw 1 From we derive hence z 1 (t) = [z 1 (0) + (γ + αβ) l 1 = αw 1 + βw 1l 1 + γw 2 1 z 1 = βw 1z 1 + (γ + αβ)w 2 1 t 0 w 2 1 (τ)e R τ 0 βw 1(s)ds dτ]er t 0 βw 1(s)ds

17 Generic assumption We shall assume that c 1 0 and that [ ] c2 b det 3 0 c 2 b 5 where c 1 = c 2 = c 2 = b 3 = b 5 = θ 1 n 1 θ 2 n 2 θ 2 y n χ 1 2 θ 2 χ 2 2 θ 1 χ 1 θ 2 y χ 2 θ 1 y

18 Main result Thm If the generic assumption holds with m >> 1, J >> 1, then the system h = Ql J l = Cw + B(l, w) with state (h, l) R 6 and control w R 2 is locally controllable around 0. The local controllability also holds true in the presence of vorticity and circulation.

19 Example 1: Elliptic boat with 3 controls Actually, the linearized system around the null trajectory is controllable!

20 Example 2: Elliptic boat with 2 longitudinal controls Generic condition fulfilled iff b 3 = 1 2 Ψ 2 n 2 0 where Ψ = 0, Ψ/ n = χ1 y2 >0

21 Step 1. Loop-shaped trajectory We consider a special trajectory of the toy problem (w 2 0) constructed as in the flatness approach due to M. Fliess, J. Levine, P. Martin, P. Rouchon We first define the trajectory h 1 (t) = λ(1 cos(2πt/t )) l 1 (t) = λ(2π/t )) sin(2πt/t ) We next solve the Cauchy problem { w1 = α 1 {l 1 γw1 2 βw 1 l 1 } w 1 (0) = 0 to design the control input. Then w 1 exists on [0, T ] for 0 < λ << 1. (h 1, l 1 ) = 0 at t = 0, T. Nothing can be said about w 1 (T ).

22 Step 2. Return Method We linearize along the above (non trivial) reference trajectory to use the nonlinear terms. We obtain a system of the form x = A(t)x + B(t)u + Cu h,l T t w T t

23 Linearization along the reference trajectory Fact. The reachable set from the origin for the system is x = A(t)x + B(t)u + Cu, x R n, u R m R = R T (A, B + AC) + CR m + Φ(T, 0)CR m where Φ(t, t 0 ) is the resolvent matrix associated with the system x = A(t)x, and R T (A, B) denotes the reachable set in time T from 0 for x = A(t)x + B(t)u, i.e. R T (A, B) = {x(t ); x = A(t)x + B(t)u, x(0) = 0, u L 2 (0, T, R m )}

24 Silverman-Meadows test of controllability Consider a C ω time-varying control system ẋ = A(t)x + B(t)u, x R n, t [0, T ], u R m. Define a sequence (M i ( )) i 0 by M 0 (t) = B(t), Then for any t 0 [0, T ] M i (t) = dm i 1 dt Φ(T, t 0 )M i (t 0 )R m = R T (A, B) i 0 A(t)M i 1 (t) i 1, t [0, T ]

25 Proof of the main result (continued) To complete the proof of the theorem in the case of potential flows, we use the generic assumption to prove that the linearized system is controllable. (We use the term w 1 w 2 to control r) the Inverse Mapping Theorem to conclude.

26 Proof of the main result (continued) In the general case (vorticity + circulation), we prove/use a Global Well-Posedness result using an extension argument (which enables us to define the vorticity at the incoming part of the flow), and Schauder Theorem in Kikuchi s spaces; Linear estimates for the difference of the velocities corresponding to potential (resp. general) flows in terms of the vorticity and circulation at time 0; a topological argument to conclude when the vorticity and the circulation are small; a scaling argument due to J.-M. Coron

27 Conclusion Local exact controllability result for a boat with a general shape Two linearization arguments: in R 6 (for potential flows) and next to deal with general flows Prospects: Motion planning 3D (submarine) (work in progress with Rodrigo Lecaros, CMM, Santiago of Chili) Numerics??